Method and System for Propagating the State of an Object and Its Uncertainty

1. A method for propagating the state of an object and an uncertainty associated with the state of the object, the method comprising: receiving, by a computer system, an interval of time over which the uncertainty is to be propagated, as well as an ensemble of initial conditions that characterize the initial uncertainty associated with the state of an object and an ordinary differential equation (ODE) describing dynamics of the state of the object, which together define an initial value problem (IVP) ensemble, for the object, the IVP ensemble comprising a plurality of IVPs; solving, by the computer system, the IVPs in the IVP ensemble wherein the solution of one or more of the IVPs in the IVP ensemble depends upon the solution of one or more of the other IVPs in the IVP ensemble; characterizing, by the computer system, the uncertainty associated with the state of the object at the end of the time interval based on solving the IVPs in the IVP ensemble.

2. The method of claim 1 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps.

3. The method of claim 1 , wherein one or more time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

4. The method of claim 1 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps, and wherein one or more the time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

5. The method of claim 1 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

6. The method of claim 2 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

7. The method of claim 3 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

8. The method of claim 4 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

9. The method of claim 5 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

10. The method of claim 6 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

11. The method of claim 7 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

12. The method of claim 8 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

13. A system comprising: a processor; and a memory coupled with the processor and having stored therein a sequence of instructions which, when executed by the processor, cause the processor to propagate the state of an object and an uncertainty associated with the state of the object by: receiving an interval of time over which the uncertainty is to be propagated, as well as an ensemble of initial conditions that characterize the initial uncertainty associated with the state of an object and an ordinary differential equation (ODE) describing dynamics of the state of the object, which together define an initial value problem (IVP) ensemble, for the object, the IVP ensemble comprising a plurality of IVPs; solving one or more of the IVPs in the IVP ensemble wherein the solution of one or more of the IVPs in the IVP ensemble depends upon the solution of one or more of the other IVPs in the IVP ensemble; and characterizing the uncertainty associated with the state of the object at the end of the time interval based on solving the IVPs in the IVP ensemble.

14. The system of claim 13 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps.

15. The system of claim 13 , wherein one or more time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

16. The system of claim 13 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps, and wherein one or more time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

17. The system of claim 13 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

18. The system of claim 14 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

19. The system of claim 15 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

20. The system of claim 16 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

21. The system of claim 17 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

22. The system of claim 18 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

23. The system of claim 19 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

24. The system of claim 20 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

25. A computer-readable memory having stored therein a sequence of instructions which, when executed by a processor, cause the processor to propagate the state of an object and an uncertainty associated with the state of the object by: receiving an interval of time over which the uncertainty is to be propagated, as well as an ensemble of initial conditions that characterize the initial uncertainty associated with the state of an object and an ordinary differential equation (ODE) describing dynamics of the state of the object, which together define an initial value problem (IVP) ensemble, for the object, the IVP ensemble comprising a plurality of IVPs; solving one or more of the IVPs in the IVP ensemble wherein the solution of one or more of the IVPs in the IVP ensemble depends upon the solution of one or more of the other IVPs in the IVP ensemble; and characterizing the uncertainty associated with the state of the object at the end of the time interval based on solving the IVPs in the IVP ensemble.

26. The computer-readable memory of claim 25 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps.

27. The computer-readable memory of claim 25 , wherein one or more time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

28. The computer-readable memory of claim 25 , wherein the solution of one or more of the IVPs, over one or more time steps, is used within the process of solving one or more of the remaining IVPs, over one or more time steps, and wherein one or more time steps taken to solve one or more of the IVPs in the IVP ensemble are the same as one or more time steps taken to solve one or more of the remaining IVPs in the IVP ensemble.

29. The computer-readable memory of claim 25 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

30. The computer-readable memory of claim 26 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

31. The computer-readable memory of claim 27 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

32. The computer-readable memory of claim 28 , wherein one or more of the numerical methods used to solve one or more of the IVPs in the IVP ensemble is an implicit Runge-Kutta method.

33. The computer-readable memory of claim 29 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

34. The computer-readable memory of claim 30 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

35. The computer-readable memory of claim 31 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.

36. The computer-readable memory of claim 32 , wherein one or more of the implicit Runge-Kutta methods is estimating and controlling the truncation error.